Results 1 to 3 of 3

Math Help - Linear operators and Differential Equations

  1. #1
    Member
    Joined
    Feb 2008
    Posts
    184

    Linear operators and Differential Equations

    Hello,

    I have missed all my lectures on the above topic due to illness. Does anyone know of a good website with solved excercise sheets to help me learn material on this topic, in particular, I am looking for helpful notes for homework questions on this website http://www.maths.qmul.ac.uk/~cchu/MTH6122/lode092.pdf

    Many Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    <L(f),g>= \int_{0} ^{1} ( \int_{0} ^{1} f(y)K(x,y)dy )g(x)dx =  \int_{0} ^{1} ( \int_{0} ^{1} g(x)f(y)K(x,y)dy )dx Using Fubini, this last one is equal to \int_{0} ^{1} ( \int_{0} ^{1} g(x)f(y)K(x,y)dx )dy = \int_{0} ^{1} ( \int_{0} ^{1} g(x)K(x,y)dx )f(y)dy = <f,L(g)> So L ^* =L ie. L is self-adjoint.

    Now suppose f is an eigenfunction of L with eigenvalue \lambda with K as defined in the sheet then:

    \lambda f(x) = \int_{0} ^{1} f(y)K(x,y)dy = \int_{0} ^{x} f(y)(1-x)ydy + \int_{x} ^{1} f(y)(1-y)xdy  = (1-x) \int_{0} ^{x} yf(y)dy + x\int_{x} ^{1} (1-y)f(y)dy. In using this last expression for \lambda f(x) it's clear that f(0)=0=f(1) (as long as \lambda \neq 0, a difficulty that'll dissapear in the next step).

    Now, assuming f is twice differentiable (only continuity will not be enough) we have:

    \lambda f'(x)=-\int_{0} ^{x} yf(y)dy +(1-x)xf(x) + \int_{x}^{1} (1-y)f(y)dy - x(1-x)f(x) = -\int_{0}^{x} yf(y)dy +\int_{x}^{1} (1-y)f(y)dy

    \lambda f''(x)=-xf(x) - (1-x)f(x)=-f(x) and so \lambda f'' +f=0. Now just notice that we never used \lambda \neq 0 so if \lambda =0 we just have f=0 which satisfies the boundary value problem trivially.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2008
    Posts
    184
    You are a life saver Jose! I will donate some money to this great site tonight.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. tangent vector fields as linear partial differential operators
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 29th 2011, 11:35 PM
  2. Linear Differential Equations
    Posted in the Algebra Forum
    Replies: 1
    Last Post: April 26th 2009, 12:43 PM
  3. Replies: 1
    Last Post: May 15th 2008, 08:23 PM
  4. linear differential equations??
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 20th 2008, 09:44 AM
  5. Replies: 5
    Last Post: July 16th 2007, 04:55 AM

Search Tags


/mathhelpforum @mathhelpforum